• Ambaum, M., , and W. Verkley, 1995: Orography in a contour dynamics model of large-scale atmospheric flow. J. Atmos. Sci., 52 , 26432662.

    • Search Google Scholar
    • Export Citation
  • Appenzeller, C., , and H. C. Davies, 1992: Structure of stratospheric intrusions into the troposphere. Nature, 358 , 570572.

  • Bell, G. I., 1990: Interaction between vortices and waves in a simple model of geophysical flow. Phys. Fluids, 2A , 575586.

  • Blackmon, M. L., , J. E. Geisler, , and E. J. Pitcher, 1983: A general circulation model study of January climate anomaly patterns associated with interannual variation of equatorial Pacific sea surface temperature. J. Atmos. Sci., 40 , 14101425.

    • Search Google Scholar
    • Export Citation
  • Branstator, G., 1985a: Analysis of general circulation model sea surface temperature anomaly simulations using a linear-model. Part I: Forced solutions. J. Atmos. Sci., 42 , 22252241.

    • Search Google Scholar
    • Export Citation
  • Branstator, G., 1985b: Analysis of general-circulation model sea-surface temperature anomaly simulations using a linear-model. Part II: Eigenanalysis. J. Atmos. Sci., 42 , 22422254.

    • Search Google Scholar
    • Export Citation
  • Branstator, G., 2002: Circumglobal teleconnections, the jet stream waveguide, and the North Atlantic Oscillation. J. Climate, 15 , 18931910.

    • Search Google Scholar
    • Export Citation
  • Davies, H. C., , and C. H. Bishop, 1994: Eady edge waves and rapid development. J. Atmos. Sci., 51 , 19301946.

  • Davies, H. C., , and A. M. Rossa, 1998: PV frontogenesis and upper-tropospheric fronts. Mon. Wea. Rev., 126 , 15281539.

  • Fehlmann, R., , and H. C. Davies, 1999: Role of salient PV-elements in an event of frontal-wave cyclogenesis. Quart. J. Roy. Meteor. Soc., 125 , 18011824.

    • Search Google Scholar
    • Export Citation
  • Majewski, D., 1991: The Europa-Modell of the Deutscher Wetterdienst. Numerical Methods in Atmospheric Models, ECMWF Seminar Proceedings, Vol. 2, European Centre for Medium–Range Weather Forecasts, 147–191.

    • Search Google Scholar
    • Export Citation
  • Massacand, A. C., , and H. C. Davies, 2001: Interannual variability of the extratropical Northern Hemisphere and the potential vorticity wave guide. Atmos. Sci. Lett.,2, doi:10.1006/asle.2001. 0027.

    • Search Google Scholar
    • Export Citation
  • McIntyre, M. E., , and T. N. Palmer, 1984: The surf zone in the stratosphere. J. Atmos. Terr. Phys., 46 , 825849.

  • Meacham, S. P., , G. R. Flierl, , and U. Send, 1990: Vortices in shear. Dyn. Atmos. Oceans, 14 , 333386.

  • Nakamura, M., , and A. R. Plumb, 1994: The effects of flow asymmetry on the direction of Rossby wave breaking. J. Atmos. Sci., 51 , 20312045.

    • Search Google Scholar
    • Export Citation
  • Palmén, E., , and C. N. Newton, 1948: A study of the mean wind and temperature distribution in the vicinity of the polar front in winter. J. Meteor., 5 , 220226.

    • Search Google Scholar
    • Export Citation
  • Pedlosky, J., 1986: Geophysical Fluid Dynamics. 2d ed. Springer-Verlag, 710 pp.

  • Platzman, G. W., 1949: The motion of barotropic disturbances in the upper troposphere. Tellus, 1 , 5364.

  • Polvani, L. M., , and R. Plumb, 1992: Rossby wave breaking, microbreaking, filamentation, and secondary vortex formation: The dynamics of a perturbed vortex. J. Atmos. Sci., 49 , 462476.

    • Search Google Scholar
    • Export Citation
  • Pomroy, H. R., , and A. J. Thorpe, 2000: The evolution and dynamical role of reduced upper-tropospheric potential vorticity in intensive observing period one of FASTEX. Mon. Wea. Rev., 128 , 18171834.

    • Search Google Scholar
    • Export Citation
  • Pratt, L. J., , J. Earles, , P. Cornillon, , and J. F. Cayula, 1991: The non-linear behaviour of varicose disturbances in a simple model of the Gulf Stream. Deep-Sea Res., 38 , (Suppl. 1),. S591S622.

    • Search Google Scholar
    • Export Citation
  • Reed, R. J., , and E. F. Danielsen, 1960: Fronts in the vicinity of the tropopause. Arch. Meteor. Geophys. Bioklimatol., 11A , 117.

  • Schwierz, C., , and H. C. Davies, 2003: Evolution of a synoptic-scale vortex advecting toward a high mountain. Tellus, 55A , 158172.

  • Simmons, A. J., , J. M. Wallace, , and G. W. Branstator, 1983: Barotropic wave propagation and instability, and atmospheric teleconnection patterns. J. Atmos. Sci., 40 , 13631392.

    • Search Google Scholar
    • Export Citation
  • Sun, D. Z., , and R. S. Lindzen, 1994: A PV view of the zonal mean distribution of temperature and wind in the extratropical troposphere. J. Atmos. Sci., 51 , 757772.

    • Search Google Scholar
    • Export Citation
  • Swanson, K. L., , P. J. Kushner, , and I. M. Held, 1997: Dynamics of barotropic storm tracks. J. Atmos. Sci., 54 , 791810.

  • View in gallery

    Instantaneous isentropic PV gradient on 1800 UTC 24 Dec 1999 (shaded, units 10−6 PVU m−1). (a) On the 320-K isentrope, 2-PVU PV contour and wind speed (contours 50, 70, 80, 90 m s−1) overlaid. (b) South–north cross section at 30°W. Vertical coordinate: θ; PV isolines [1, 2 (bold) 3, 4, 5, 6, 8 PVU]; and 50, 70, 80, 90 m s−1 isotachs (dashed) overlaid

  • View in gallery

    Mean winter (DJF) isentropic PV gradient for the 15-yr period 1979–93 (ERA-15 data). (a) On 320 K with wind speed (contours 30, 40, 50 m s−1). (b) Cross section at 75°W; 30 and 40 m s−1 isotachs (dashed) overlaid. Same as Fig. 1, but note the difference in scale and magnitude

  • View in gallery

    (a) Instantaneous PV on the 320-K isentrope at 1800 UTC 24 Dec 1999 (shaded, in PVU) with 2-PVU PV contour (bold black) and synoptic and mesoscale vortex anomalies (labels A–F) overlaid; (b) 15-yr winter (DJF) mean of negative vorticity on 500 hPa (shaded, units 10−4 s−1)

  • View in gallery

    Schematic diagram to illustrate the analytic model setup, terms, and symbols. (top) A vertical section at y = d. See text for details.

  • View in gallery

    (a) Initial velocity υa (dotted) and perturbation velocities after 2 (light) and 5 days (bold); units along x axis are 1000 km. (b) Amplitude evolution for different modes k in the domain of length Lϕ = 2πR cosϕ, with R (earth's radius) and Δk = 2π/Lϕ: 3Δk (dotted), Δk (bold) and 5Δk (light). Time in days. (c) Energy evolution [normalized, E(t)/E(0)] for the two extreme cases kr = 3Δk (bold) and kr = 3.5Δk (light). Also shown (dotted) is a function proportional to t2 (for t ≥ 10 days). (d) Spatial flow pattern after 5 days

  • View in gallery

    (a) Zonal velocity profile (solid line, units m s−1) and vorticity discontinuity [dashed, values 7.25 and 15 (×10−5 s−1)] of the barotropic initial state for the PE model operated on the sphere. The PV discontinuity is located at 47.5°N. (b) Initial positive vorticity anomaly (shaded, units 10−4 s−1) located north of the PV discontinuity (solid line).

  • View in gallery

    Time evolution (60–300 h) of the perturbation vorticity response (shaded, units 10−4 s−1) for a moving vorticity anomaly acting on a vorticity gradient at 47.5°N. Latitude circles shown for 40° and 80°N.

  • View in gallery

    Perturbation vorticity response at wavenumbers (top) m = 5, (bottom) m = 6, and (middle) for a nonresonant case for the integration times (left) 240 h and (right) 315 h. The different wavenumbers result from the particular location of the anomaly (bold contours 0.1, 0.3, 0.5 PVU) relative to the waveguide (65.5°, 63°, and 61.5°N, respectively). Cylindrical projection of the region 40°–75°N, 180°W–180°E.

  • View in gallery

    (a) Positive and (b) negative synoptic-scale vortex anomalies (a0 ∼ 700 km, lat0 ∼ 61°N) acting on a discontinuity at 47.5°N. Perturbation isentropic PV on 320 K for 0, 24, 36, 48, and 72 h. Contour interval is 0.15 PVU. Vortex anomaly amplitudes at initial time are 12 and −1.5 PVU, respectively

  • View in gallery

    Time evolution (60–300 h) of the perturbation vorticity response (shaded, units 10−4 s−1) for both an orographic stationary vorticity source (dashed contours) and a moving vorticity anomaly (solid contours) acting on a vorticity gradient at 47.5°N.

  • View in gallery

    Vorticity Hovmöller diagram at 47.5°N for a 480-h PE simulation (PV discontinuity located at 47.5°N, orography centered at 45°W)

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 135 135 8
PDF Downloads 18 18 3

Forced Waves on a Zonally Aligned Jet Stream

View More View Less
  • 1 Institute for Atmospheric and Climate Science, ETH, Zürich, Switzerland
© Get Permissions
Full access

Abstract

The potential vorticity (PV) pattern in the vicinity of the jet stream takes the form of a narrow tube of enhanced PV gradient on the in situ isentropic surfaces. It is asserted that this distinctive structure can serve as a waveguide and a seat for trapped Rossby waves and that a neighboring vortexlike anomaly can trigger such waves and/or interact strongly with the jet. These conjectures are examined theoretically in an idealized setting comprising a finite-scale vortex forcing of a zonally aligned PV discontinuity. The quintessential dynamics of the vortex's influence upon the PV interface are first elucidated in the linear barotropic β-plane limit, and thereafter other aspects of the jet–vortex interaction are examined in a hemispheric primitive equation setting using a nonlinear numerical model.

It is shown that for the selected setting the interface can sustain trapped waves, a strong response is favored by larger-scale forcing, and a quasi-resonant response can prevail for some ambient flow settings, provided the vortex advects zonally at approximately the Doppler-shifted velocity of a trapped Rossby wave. It is also deduced that (i) a mesoscale perturbing vortex can retain its coherency despite the deforming effect of the ambient flow; (ii) the enhanced PV gradient can indeed serve as an effective waveguide; and (iii) the backreaction of the interface perturbations upon a weak mesoscale vortex need not be appreciable, and conversely for a stronger synoptic-scale vortex the interaction can lead to significant deformation of both vortex and interface with a tendency for a pairing of the vortex with an oppositely signed anomaly on the distorted interface. Comments are made on the relationship of the results to observed phenomena.

Corresponding author address: Cornelia Schwierz, Institute for Atmospheric and Climate Science, ETH Hönggerberg, CH-8093 Zürich, Switzerland. Email: schwierz@iac.umnw.ethz.ch

Abstract

The potential vorticity (PV) pattern in the vicinity of the jet stream takes the form of a narrow tube of enhanced PV gradient on the in situ isentropic surfaces. It is asserted that this distinctive structure can serve as a waveguide and a seat for trapped Rossby waves and that a neighboring vortexlike anomaly can trigger such waves and/or interact strongly with the jet. These conjectures are examined theoretically in an idealized setting comprising a finite-scale vortex forcing of a zonally aligned PV discontinuity. The quintessential dynamics of the vortex's influence upon the PV interface are first elucidated in the linear barotropic β-plane limit, and thereafter other aspects of the jet–vortex interaction are examined in a hemispheric primitive equation setting using a nonlinear numerical model.

It is shown that for the selected setting the interface can sustain trapped waves, a strong response is favored by larger-scale forcing, and a quasi-resonant response can prevail for some ambient flow settings, provided the vortex advects zonally at approximately the Doppler-shifted velocity of a trapped Rossby wave. It is also deduced that (i) a mesoscale perturbing vortex can retain its coherency despite the deforming effect of the ambient flow; (ii) the enhanced PV gradient can indeed serve as an effective waveguide; and (iii) the backreaction of the interface perturbations upon a weak mesoscale vortex need not be appreciable, and conversely for a stronger synoptic-scale vortex the interaction can lead to significant deformation of both vortex and interface with a tendency for a pairing of the vortex with an oppositely signed anomaly on the distorted interface. Comments are made on the relationship of the results to observed phenomena.

Corresponding author address: Cornelia Schwierz, Institute for Atmospheric and Climate Science, ETH Hönggerberg, CH-8093 Zürich, Switzerland. Email: schwierz@iac.umnw.ethz.ch

1. Introduction

A striking and ubiquitous feature of atmospheric flow is a meandering jet stream that is located at a break in the tropopause and almost girdles the globe. A related feature is the distinctive structure of the accompanying potential vorticity (PV) distribution. It takes the form of a band of enhanced PV gradient aligned along the jet and located on isentropic surfaces that transect the tropopause break. This band of enhanced PV gradient (∇θPV) constitutes the starting point of the present study.

Prototypical examples of both the instantaneous and the climatological structure of the band are shown in Figs. 1 and 2. These depictions are in line with earlier perspicacious analyses of these features (e.g., Palmén and Newton 1948; Reed and Danielsen 1960), and serve to emphasize the coalignment of the band with the jet, the weak PV gradient of the ambient atmosphere, and the strength of the band itself. Coalignment conforms with the following quasigeostrophic relationship (cast in scaled isentropic coordinates):
θ2θU,
where U denotes the flow component directed along the jet axis. In effect the relationship indicates that a local maximum of ∇θPV also connotes a maximum in the flow. Homogeneity of the ambient tropospheric PV field accords with the mixing of PV by transient eddies (e.g., McIntyre and Palmer 1984; Sun and Lindzen 1994), and likewise the enhanced gradient accords with PV frontogenesis at the tropopause break accompanying the evolution of these eddies (Davies and Rossa 1998). Note also that the amplitudes of the instantaneous and the time-mean ∇θPV bands amount, respectively, to ∼50 × 10−6 pvu m−1 and ∼3 × 10−6 PVU m−1 (PVU: PV unit) and these values correspond to ∼O(100) and ∼O(10) enhancement of their ambient atmospheric or the purely β-attributable gradient.

Irrespective of a band's origin its existence, structure, and amplitude carry important dynamical ramifications. First, Rossby waves propagate on the ∇θPV field, and the latter field's localized structure exerts a direct influence upon the dynamical properties and transmissivity of the waves. Second, the strength of the two types of bands imply that large-scale adiabatic displacement of air parcels across the band would produce PV anomalies of ∼7 PVU and ∼3 PVU, respectively, and in turn such anomalies would connote significant modifications of the in situ (and far field) flow and thermal distributions.

These inferences suggest that the band's structure, allied to its temporal persistence and streamwise length, marks it out as an important tropopause-level dynamical feature and a possible waveguide in a variety of flow settings. Two such settings are (i) the transient response to and interaction with a juxtaposed meso- or synoptic-scale vortexlike anomaly and (ii) the quasi-steady response to lower-level planetary-scale stationary forcing.

The motivation and rationale for considering the first setting arise from the ubiquity and amplitude of such anomalies. For example Fig. 3a, which shows the PV distribution on the 320-K surface for the same instant as Fig. 1, pinpoints the presence of meso- and synoptic-scale positive PV anomalies (features A, B, and C) and a negative anomaly (E) that are all in the vicinity of the ∇θPV band. [Some anomalies (D and F) are removed from the band]. In contrast, for the second setting the effect of the low-level orographic forcing would only be evident at higher elevations as an anticyclonic vorticity pattern, and there is a hint of such a signal in Fig. 3a over the northern part of the Rockies. A more vivid depiction of this orographic signal is evident in the time-mean vorticity field of Fig. 3b, which shows that the flow over the Rockies, Greenland, and the Himalayas is characterized by negative vorticity values that are almost unrivalled elsewhere in the extratropics.

The present heuristic study is designed to explore aspects of the ∇θPV band's influence in the foregoing two settings. The approach adopted is based upon first representing the band in the highly idealized form of a latitudinally and vertically aligned interface separating two regions of uniform (but different) potential vorticity. Then the response is evaluated when the interface is perturbed by an isolated vortex that is either fixed in space or advects with the ambient flow.

This idealized configuration has a long pedigree in atmospheric studies (Platzman 1949) and has been invoked in several recent studies (Polvani and Plumb 1992; Nakamura and Plumb 1994; Ambaum and Verkley 1995; Swanson et al. 1997; Pomroy and Thorpe 2000). It has the merit of restricting Eulerian in situ adiabatic PV changes to the neighborhood of the interface and the vortex and, moreover, in a simple two-dimensional setting it has been shown that the interface can sustain trapped Rossby waves. The configuration has also been adopted in oceanographic studies to model strong currents such as the Gulf Stream and such a current's meanders and interaction with an isolated vortex (see Bell 1990; Pratt et al. 1991 and references therein). In particular it has been shown (Bell 1990) that a point vortex can excite a resonant response of the trapped wave, and that a backreaction of perturbations on the interface induces a “latitudinal” displacement of the point vortex.

The paper is organized as follows. In section 2 the quintessential dynamics of a finite-scale vortex's influence upon the interface is examined in a simple linear β-plane setting. This setting is designed to complement, elaborate, and extend the results of earlier studies (in particular that of Bell 1990) and to concomitantly provide a transparent illustration and interpretation of the linear ingredients of the vortex's influence. In section 3 the nature of this influence is tested in the more realistic but still idealized atmospheric setting of nonlinear hemispheric primitive equation flow. This setup permits explicit examination of the jet–vortex interaction for vortices of different spatial scale, amplitude, and origin. Finally, in section 4 the derived results are placed in the context of extant studies of jet–vortex interaction and quasi-steady forcing of planetary- and subplanetary-scale atmospheric waves.

2. Interfacial waves on a β plane

a. The flow setting

Quasigeostrophic flow of a homogeneous fluid on a β plane with bottom topography η = η(x, y) and a rigid upper lid at a height H (see Fig. 4) satisfies the following potential vorticity equation
i1520-0469-61-1-73-e1
Here q = {ξ + (f0/H)η + βy} is the potential vorticity, Dg/Dt is a pseudo-Lagrangian derivative following the geostrophic flow, ξ the vorticity, and ψ denotes the corresponding geostrophic streamfunction (so that ξ = ∇2ψ). Pedlosky (1986) provides a resume of this flow setting.
A basic state is selected to comprise a zonal flow over flat terrain with (as stated earlier) a zonally aligned interface located along y = 0 separating two semi-infinite domains of uniform but different values of quasigeostrophic potential vorticity, q = ±P/2. The corresponding zonal flow field is given by
i1520-0469-61-1-73-e2
with sgn = sign(q). Small perturbations of this basic state satisfy a linearized form of Eq. (1):
i1520-0469-61-1-73-e3
where υ′ denotes the perturbation meridional velocity field and q′ is the perturbation potential vorticity given by
i1520-0469-61-1-73-e4

b. Free waves

In the absence of topography the foregoing setting is standard, and we briefly recapitulate previous results. Wave perturbations exist with the streamfunction in the two domains taking the form
i1520-0469-61-1-73-eq2
with (ω, k) denoting, respectively, the frequency and the zonal wavenumber. The dispersion relationship is obtained by integrating Eq. (3) across the PV interface, and is given by (see also Bell 1990; Swanson et al. 1997)
i1520-0469-61-1-73-e5
Thus, the natural modes of the system are trapped on the interface and their latitudinal decay scale is determined by the zonal wavenumber k. Consistent with the underlying Rossby wave character (i.e., propagation on a PV gradient) the individual modes propagate westward relative to the basic-state velocity field U0 at the interface. However, in contrast to Rossby waves on a uniform flow, the zonal group velocity of these waves is given by U0 and is independent of k. Also, note that the dispersion relationship depends exclusively upon the in situ velocity and the vorticity jump at the interface, and not upon the other details of the background basic state.
Note further that the steady-state wave is characterized by the wavenumber ks such that
i1520-0469-61-1-73-e6
Note that for U0 ∼ 30 m s−1 and mean basic-state vorticity values (P/2) in the broad range (3–6) × 10−5 s−1, the value of ks equates in spherical geometry to a zonal wavenumber m in the band of m ∼ 4–8. [A comparable relationship to Eq. (6) is derived in section 3a for the spherical geometry configuration.]
Thus, the simple two-zone setting can sustain dynamically trapped Rossby waves, and moreover their existence and lateral confinement implies that a forcing in the far field could excite (and possibly resonate with) a trapped wave on the interface. For a stationary forcing field, resonance would be expected to prevail for a wavenumber k satisfying Eq. (6). Alternatively for a forcing field advecting steadily eastward with a velocity U* the corresponding wavelength is given by
i1520-0469-61-1-73-e7
In the next section we examine and illustrate the dynamics of a setup with a stationary vortexlike forcing field.

c. Isolated topographic vortex forcing

Consider now the setting with the prescribed basic flow state [i.e., satisfying Eq. (2)] perturbed by traversing over an isolated topographic feature. In effect this setup represents the forcing of waves on the interface due to a spatially fixed topographically forced vortex. The overall dynamics is now governed by Eq. (3), and the perturbation streamfunction ψ′ satisfies the relationship [see Eq. (4)]
i1520-0469-61-1-73-eq3
It follows that the streamfunction is composed of two components (say, ψa and ψb), representing respectively a (possibly) time-dependent contribution due to perturbations of the interface (the q′ term) and time-independent topographically induced component (the η term).
Consider in turn the form of these two components. The first component can be written as a Fourier series of waves, each of the form
i1520-0469-61-1-73-e8
with (a, ε) denoting the time-dependent amplitude and phase of the kth wavenumber.
For the second (η) term consider an axisymmetric mountain of radius R located at a distance d (d > R) from the PV interface (see Fig. 4). The mountain-induced circumferential velocity V(r) at a radius r takes the form
i1520-0469-61-1-73-e9
where K is a measure of the orographically induced vortex strength such that
i1520-0469-61-1-73-eq4
for a top-hat-shaped mountain of height η0. The associated perturbation meridional velocity at the interface can be written successively (using of the Fourier sine transform) as
i1520-0469-61-1-73-eq5
Thus the contribution of the kth wavenumber to the meridional velocity at the interface can be written in the form
i1520-0469-61-1-73-e10
Insertion of ψ′, derived from using Eqs. (8) and (10), into Eq. (3) followed by integration across the interface at y = 0 yields the following coupled set of equations for the time evolution of the kth wave component (cf. Davies and Bishop 1994):
i1520-0469-61-1-73-e11
with
i1520-0469-61-1-73-e12
The first equation (11) prescribes the growth of the interfacial wave due to the vortex forcing. Growth is favored by a strong PV jump at the interface, a small wavenumber, proximity of the vortex to the interface, and an appropriate phase alignment (the optimum prevails for εk = −π/2). The second equation (11) indicates that the net phase change arises from the difference between the westward propagation of a free mode (the γ term) and the vortex-induced change. We note in passing that the nonlinear feedback due to the interfacial waves acting upon the topographically induced component is excluded in the linearized system represented by Eq. (3), but we return to consider this aspect in section 3.

It is instructive to consider two limit forms of the coupled set.

(i) A time-independent response:
In this case each wavenumber is prescribed by
i1520-0469-61-1-73-eq6
Thus, a steady state prevails when the westward propagation of the Rossby wave trapped on the interface is matched by an opposing tendency attributable to the vortex forcing (see also Bell 1990).
(ii) The resonant response:
This response prevails for a wavenumber satisfying the relationships
kπγ
so that
aktt.
These criteria equate to the optimum configuration of the vortex and wave for the latter's growth (i.e., εk = −π/2), and it is sustained because the corresponding free mode is stationary relative to the forcing (γ = 0, cosεk = 0).
A more general solution of the coupled set [Eqs. (11)] can be found on noting that the set possesses the following integral invariant:
i1520-0469-61-1-73-e13
Hence the evolution for an initially unperturbed interface, ak(0) = 0, is given by
i1520-0469-61-1-73-e14
with the accompanying phase evolution given by
kμak
It follows from Eqs. (8), (14), and (15) that the ψa component of the streamfunction can be rewritten as
i1520-0469-61-1-73-e16
This neat two-term formulation of the waveguide's contribution was pointed out to us by a reviewer. The first term is stationary and is a direct modification of the vortex's contribution to the streamfunction, and the second is a propagating wave whose instantaneous velocity is given by (−2ε/k).

Illustrations of the flow evolution for an elliptic-shaped mountain centered 1000 km poleward of the interface and with a major axis of 1000 km are provided in Fig. 5. (Note also that H = 10 km, P = 5 × 10−5 s−1, h0 = 2 km, and a β plane with an east–west span of 27 000 km.) Figures 5a–c show the temporal development of the flow, and Fig. 5d the spatial pattern after 5 days. These depictions show a train of interfacial waves that evolve downstream with the ambient flow (sic. the group velocity), and their wavelength (∼8400 km) matches that of the stationary wave [Eq. (6)]. The wave train is well developed after 5 days, and after some 10 days the wave train approaches the “longitude” of the topography from the west.

Thereafter there is evidence of constructive interference. Clearly the nature of the interference—constructive or destructive—depends crucially on whether the β plane's east–west span equates to an integer multiple of the stationary wavelength. This behavior is illustrated (see Fig. 5c) by examining the time trace of normalized wave energy on the interface, E(t)/E(0), where
i1520-0469-61-1-73-eq9
for parameter settings (P, U0) corresponding to a potentially resonant (γ = 0) setting and a nonresonant setting. Both experiments show a similar short-time evolution, but the longer-time development is markedly different. The energy for the “resonant” mode grows like t2 in the longer-time limit, whereas the amplitude of the off-resonant mode merely oscillates with time.

3. Interfacial waves on a hemisphere

The analysis of the previous section does not account for effects beyond β-plane geometry, quasigeostrophy, linearity, and a discontinuous PV interface. In this section we consider first the form of free waves on a barotropic PV discontinuity in spherical geometry and then examine the robustness of the dynamics using a nonlinear primitive equation model with a finite-width PV transition zone.

a. Free waves

Consider the analogue of the flow setting of section 2b but now in a hemispheric geometry. The basic-state absolute vorticity is set to a value of P poleward and zero equatorward of a specified latitude ϕ0. The corresponding zonal velocity U expressed in terms of U = U cosϕ is then given by
i1520-0469-61-1-73-eq10
with a and Ω denoting the radius and angular velocity of the earth. Assuming a zero-perturbation meridional velocity at the equator, then the streamfunction in the two domains is given by
i1520-0469-61-1-73-eq11
Here (λ, m) refer, respectively, to longitude and the zonal wavenumber, p = tanh−1 (sinϕ), and the amplitude modifying factor Γ is such that Γ = −emp0/(2 sinhmp0).
The dispersion relationship takes the form
i1520-0469-61-1-73-e17
where Ω0 is the angular velocity of the zonal flow at the latitude of the PV interface.
The correspondence with the earlier β-plane dispersion relationship is brought out on noting that for an interface located in midlatitudes
mp0
provided m ≥ 3. In this limit the waves are in effect latitudinally trapped to the interface, the dispersion relationship reduces to
i1520-0469-61-1-73-e18
and the azimuthal group velocity ∂ω/∂m is approximated by Ω0; that is, it is determined by the angular velocity at the interface.
In the same limit the equivalent for spherical geometry (with m ≥ 3) of the resonant waves prescribed by Eqs. (6) and (7) are given by
i1520-0469-61-1-73-e19
Here Eqs. (19), (20) apply, respectively, to a stationary setting and a forcing field that advects eastward with an angular velocity Ω*0. Thus, in spherical geometry resonance can prevail if the flow setting is such that the ratio of P and either Ω0 or (Ω0Ω*0) deliver an integral number corresponding to the zonal wavenumber m. In this context note that P and Ω0 are not interdependent since
P00ϕ0
or equivalently
P0−1ϕ0−1P

b. Primitive equation simulations

Idealized numerical simulations are performed in a primitive equation (PE) framework to examine jet–vortex interaction on the hemisphere. The basic-state setting resembles that used in section 2.

1) Model setup

Simulations are undertaken with an adiabatic version of the so-called Europa Model (EM) of the German Weather Service (Majewski 1991). This gridpoint model is used herein in a pole-centered, rotated-grid configuration on a quasi-hemispheric domain and operated with a 1° (∼100 km) resolution, 27 vertical levels, a free-slip lower boundary condition, a rigid upper-lid, weak fourth-order horizontal diffusion, and a relaxation zone in the equatorial region.

2) Basic state

One component of the basic state in the extratropics is a barotropic zonal flow associated with two regions of uniform PV separated by a zonally aligned transition zone that extends over some 4° of latitude in the form of a column of strong horizontal PV gradient. The structure of the resulting barotropic jet and the accompanying vorticity distribution are shown in Fig. 6. This initial configuration bears comparison with the atmospheric settings discussed earlier that were portrayed in Figs. 14.

The other component comprises a vortexlike structure that takes one of the following three observationally motivated forms: (i) a single interior positive vorticity anomaly of meso-α scale (radius a0 ∼ 300 km) embedded within the high PV region (Fig. 6b) that is free to advect with the ambient flow and located at tropopause elevations at various distances from the transition zone, (ii) a counterpart positive or negative anomaly of synoptic-scale (a0 ∼ 700 km) (Fig. 9), and finally (iii) a hybrid setup with both an orographically bound negative anomaly and an advecting surface-based positive thermal anomaly.

These three configurations relate to different physical settings that were alluded to in the introduction (see also Fig. 3). Consider first the setting in case I with a positive vortex located poleward of and adjacent to a jet stream (cf. features marked A, B, and C in Fig. 3a). Both these ingredients have been implicated in the surface cyclogenesis. Incipient wavelike meanders of the jet stream (sic. ∇θPV waveguide) are often taken to be indicative of the upper-level signature of a troposphere-spanning growing baroclinic wave system (sic. baroclinic instability), and likewise a tropopause-level positive PV anomaly is noted as a possible precursor of surface frontal wave development. In effect these two categories bear comparison, respectively, with so-called Types A and B events of cyclogenesis.

In the present setup, consideration of interlevel interaction is eschewed and the focus was on single-level interaction. This is motivated by the recognition that frequent close proximity of a tropopause-level positive PV anomaly with the jet stream suggests that their interaction can indeed be a significant if not a dominant feature of the short-term development (see, e.g., Fehlmann and Davies 1999).

The setting in case II with a negative synoptic-scale vortex anomaly located within the high PV domain corresponds to the not-infrequent atmospheric setting when a large-scale upper-tropospheric air mass associated with a strong high pressure system has been sequestered into the stratosphere (cf. features marked D and E in Fig. 3a). The reverse can also occur with sequestration of stratospheric air into the upper troposphere in the form of an elongated PV streamer that can break up into a compact vortexlike structure (Appenzeller and Davies 1992). This is illustrated by the feature marked F in Fig. 3a.

Included in case III is the ingredient of perturbations on the ∇θPV band induced by a quasi-steady anticyclonic vorticity above topography (cf. the negative vorticity over the Rockies and Greenland in Fig. 3b). Other possible sources of such a steady forcing field would be regions of anomalous diabatic heating associated with SST anomalies in the subtropics. Again for a strong response the amplitude and scale of the forcing need to be appreciable and sustained, and concomitantly the ∇θPV band needs to be coherent. More trenchantly, on the longer time scale the possibility of a resonant response cannot be excluded a priori.

3) Integral diagnostic

For diagnostic purposes it is useful to note that inviscid barotropic flow on the hemisphere possesses the following integral invariant:
i1520-0469-61-1-73-e21
This invariant has direct ramifications for our particular setting of two zones of uniform PV and a single unconstrained vortex. It follows that growth of wave perturbations beyond the transition zone would need to be countered by a compensating latitudinal movement of the vortex (cf. Bell's backreaction). Thus, for example a large amplitude wave pattern on the interface would be accompanied by the movement of a positive (negative) vortex toward (away) from the interface.

A description and discussion is now provided of the simulated patterns.

4) Case I: Mesoscale vortex forcing

An interior, meso-α scale, positive PV anomaly is located some 14° poleward of a PV interface at 47.5°N. For this setting the component of the meridional velocity within the transition zone attributable to the vortex forcing is comparatively weak and the response should be at most quasilinear.

Figure 7 shows the perturbation vorticity field on the 500-hPa surface for the period 60 to 300 hours. In accord with linear theory a trapped wave train of PV anomalies develops within the transition zone of high PV gradient. It evolves downstream with the background in situ flow speed, and its wavenumber (m = 6) conforms to that expected from Eq. (20).

Consider the amplitude and structure of the induced patterns. First, note that the wave train is confined to the near neighborhood of the PV transition zone. In effect, nonlinear effects are small and the perturbation energy is exported downstream rather than being available ab initio to build up the in situ perturbation amplitude. Indeed the wave train's vorticity anomalies are ∼0.25 × 10−4 s−1 and this amounts to only a third of the vorticity difference across the transition zone.

Second, note that the backreaction (Bell 1990) of the wave train upon the vortex's latitudinal movement appears to be weak. Indeed the vortex advects eastward at a quasi-constant latitude in the time period beyond 60 hours. From a synoptic-dynamic standpoint it is evident that the prevailing configuration is such that the isolated vortex is located only slightly to the west of the first (negative) vorticity perturbation on the wave train, and concomitantly the meridional velocity at the vortex center is comparatively weak (not shown). In effect, the realized phase is not supportive of a strong backreaction of the wave train upon the vortex's movement. From a global-integrative standpoint further insight on the weak backreaction can be gleaned from Eq. (21). The wave train's weak amplitude and its confinement to the vicinity of the transition zone implies that the lateral movement of the vortex required by Eq. (21) does not need to be appreciable.

Third, the isolated vortex retains a striking coherence throughout the simulation despite the latitudinal shear of the ambient flow. This is another demonstration that nonlinear dynamical effects can enable a vortex to counter the deforming effects of an ambient shear (Meacham et al. 1990).

Fourth, the wave train encircles the globe after 285 h and this is followed by a measure of constructive interference with the m = 6 pattern persisting at least until 480 h (not shown). This longer-term behavior is the manifestation of the resonant buildup of the wave train's amplitude and accords with the linear theory of the previous section. However, it also points to the theory's limited validity and applicability. A temporal buildup of the wave amplitude also requires [see Eq. (21)] a corresponding movement of the vortex away from the critical latitude [defined by Eq. (20)], and this implies that the longer-term response can at most be a nonsingular resonance.

The sensitivity of the nature of the response to the vortex's latitude [cf. Eq. (20)] is brought out by conducting three simulations with the vortex placed at different initial latitudes corresponding approximately to constructive interference at zonal wavenumbers of m = 5 and 6, and an intermediate destructive setting. Figure 8 shows the perturbation vorticity pattern on the 500-hPa surface for each simulation at a time instance (t = 240 h) shortly before the wave train has encircled the globe and at a later instant (t = 315 h). It is evident that destructive interference prevails downstream in the off-resonant setting.

5) Case II: Synoptic-scale vortex forcing

Counterpart simulations to case I are performed with first a positive and then a negative synoptic-scale vortex (a0 ∼ 700 km). The scale and strength of the anomalies betoken significantly larger initial velocity field at the interface that is attributable to the vortex, and thereby the likelihood of nonlinear effects influencing the subsequent flow evolution. Figures 9a and 9b show, for the positive and negative anomalies respectively, the perturbation PV fields at tropopause elevations for the initial time and at 24, 36, 48, and 72 h.

The flow evolution differs markedly for the two cases, but there are three common underlying dynamical effects. First, both vortices undergo a strong asymmetric deformation under the influence of the ambient flow's lateral shear while perturbations evolve on the interface. During this phase the more equatorward portion of the vortex becomes distended longitudinally to form an elongated filament.

Second, the distorted vortex's own velocity signal induces in the case of the positive (negative) anomaly a poleward (equatorward) displacement of the filament away from (toward) the PV interface and thereby serves to weaken (enhance) the filament's interaction with anomalies evolving on the interface. The sign of the displacement is in accord with Eq. (21).

Third, there is a tendency for the vortex to become aligned with an adjacent oppositely signed anomaly on the interface whereas, in contrast, the filament becomes juxtaposed to or coalesces with a similarly signed interface anomaly.

In the subsequent evolution (not shown) the positive vortex sheds its filament, regains its circular shape at a more poleward latitude, and thereafter a wave train develops on the interface as in case I. In contrast the negative anomaly approaches the interface and eventually is smeared out. Thus the finite amplitude developments in the two cases are radically different but consistent with the integral invariant [Eq. (21)].

6) Case III: Hybrid vortex forcing

In this third case Greenland-scale and height topography is located poleward of the interface. The initial conditions are such that the perturbing effect of the topography induces two vortexlike structures that subsequently proceed to influence the PV transition zone. One vortex is the topographically bound anticyclone (cf. the forcing discussed for the β-plane setting of section 2). In effect this vortex results from the fission of the incident potential vorticity into a negative relative vorticity compensated by enhanced thermal stratification as the flow surmounts the terrain (cf. Schwierz and Davies 2003).

The second vortex is shed off the topography in the first phase of the simulation and advects with the ambient flow. It is a surface-based positive thermal anomaly resulting from the imposed initial conditions of uniform stratification and possesses a concomitant positive vorticity.

In this case (Fig. 10) a composite pattern develops with a short-wavelength wave train (m ∼ 6) evolving downstream of the advecting vortex and a longer-wavelength and weaker-amplitude wave train (m ∼ 3) evolving downstream from the stationary topographically bound anticyclone. The difference in wavelength is in accord with Eqs. (19) and (20) with the smaller relative velocity of the advecting vortex associated with a shorter-wavelength wave train. The difference in vortical amplitude accords with the reduction of the energy flux transferred to the stationary wave train.

An alternative depiction of the development is shown in Fig. 11 in the form of a Hovmöller diagram. The development occurs in three phases. An initial development of the longer-wavelength features followed by the codevelopment of the almost spatially separate wave trains, and finally the emergence of an interference pattern (with m ∼ 5) after the short-wavelength wave train approaches the longitude of the topography.

7) Further remarks

In case I the simulations with the mesoscale vortex indicates that a major distortion of the jet conducive to deep cyclogenesis requires stronger vortex forcing. Such forcing with concomitant in situ development, rather than energy propagation downstream along the waveguide, could ensue [see Eq. (21)] with a pairing of the positive anomaly and a negative anomaly created on the interface and their subsequent poleward displacement (cf. case II).

In case II the simulations involved strong nonlinear effects and there was a tendency for the negative vortex to move toward, and conceivably for a larger amplitude vortex to be extruded into the low PV domain. Such an expulsion is often observed on isentropic PV charts and the result has a bearing upon the assessment of net stratosphere–troposphere exchange of mass and chemical constituents.

For case III the simulation showed that the ∇θPV band can indeed serve as a planetary-scale waveguide transmitting the effects of the forcing to the far field along a well-defined and confined path. Note on the other hand (cf. case II) that a large amplitude forcing can deform the band irreversibly and thereby hinder propagation downstream.

4. Final remarks

The present study was predicated upon the assertion that the distinctive structure of the PV pattern in the neighborhood of the tropopause-level jet stream can exert a major influence upon large-scale atmospheric flow. In particular, it was conjectured that the accompanying band of highly enhanced ∇θPV can serve as an effective waveguide for large-scale atmospheric flow and the seat for trapped Rossby waves forced by juxtaposed isolated PV anomalies and/or larger-scale orography.

These conjectures were tested in an idealized setting comprising a vortexlike forcing of a zonally aligned PV discontinuity. First, the quintessential dynamics of the vortex's influence upon the PV interface were elucidated in the linear barotropic β-plane limit. In this setting trapped Rossby waves can be sustained on the interface, and our analysis pointed to the factors required to produce a large response to the forcing. These included planetary- or subplanetary-scale steady-state forcing or synoptic-scale forcing for an advecting vortex, the possible establishment of a resonant response in the longer-time frame for a suitable large-scale flow setting and a vortex moving zonally at the Doppler-shifted velocity of a trapped Rossby wave.

Thereafter further aspects of the interaction were examined in a hemispheric primitive equation setting using a nonlinear numerical model. Simulations performed with various forcing configurations served to lend credence to nonsingular resonant interaction in a setting with weak forcing and to indicate the disparate range of development patterns in a setting with stronger forcing.

Clearly the generality and applicability of the derived results is limited by the selection of a simple idealized setting. In reality, the domain of enhanced PV gradient on isentropic surfaces is a narrow tube that meanders around the globe and is often breached by large-amplitude breaking waves rather than taking the form of a zonally and vertically aligned interface (see Figs. 1 and 3). Moreover, the presence of both the tropopause-level tube of enhance PV gradient plus the narrow zone of enhanced surface baroclinicity (excluded herein) constitute two waveguides and, as in classical baroclinic instability, perturbations on these waveguides can interact to their mutual enhancement.

Nevertheless the derived results do bear comparison with and shed light on observed phenomena. To set the results in context it is appropriate to note that the present study's spatially confined Rossby waves are trapped internally by the atmosphere's dynamical (PV) structure and therefore differ intrinsically from classical Rossby waves propagating on comparatively smooth PV fields. Moreover, the markedly weak isentropic PV gradients away from the jet limits the efficacy of meridional wave propagation and the concomitant energy flux of conventional Rossby waves.

For synoptic-scale flow one inference is that the initiation of major upper-level wave development would be favored by a large amplitude, suitably scaled, and latitudinally located PV anomaly. The latitude corresponds to a location where the vortex would advect eastward with a zonal velocity matching that of the trapped wave. It is pertinent to record that such anomalies are more likely to be located at the tropopause level. Again, subplanetary-scale (m ∼ 3–7) steady-state forcing could be effective for a suitable large-scale flow setting. In this case an informative and illustrative depiction of the markedly different nature of wave propagation in superrotational flow versus one with a localized jet structure is shown in Branstator (1985a, Fig. 3). This result points to the efficacy of the ∇θPV band's role as a waveguide for subplanetary–scale waves. The propagation on a PV waveguide is also in accord with simulations performed with a barotropic zonally varying basic state derived from the climatological Northern Hemisphere 300-hPa field (Blackmon et al. 1983; Branstator 1985b, 2002) and with the more regularly structured waves observed on the Southern Hemisphere jet (Simmons et al. 1983). In a related vein it is pertinent to note that the subplanetary scale carries most of the signal of the atmosphere's interannual variability, and Massacand and Davies (2001) interpreted the observed patterns of this variability directly in terms of wave propagation along the time-mean ∇θPV band—in effect the PV waveguide. Hence, the analysis outlined here lends further support to a complementary interpretation of the establishment of teleconnection patterns.

Acknowledgments

The authors would like to express their gratitude to René Fehlmann for providing the PV inversion routine and support with an early version of the simulation setup and to Daniel Lüthi for expert help with the EM model. The neat deduction leading to Eq. (16) was pointed out to us by one of the reviewers. We thank ECMWF and MeteoSwiss for providing access to the meteorological data. This study was conducted in part with funding from the NCCR Climate program of the Swiss National Science Foundation.

REFERENCES

  • Ambaum, M., , and W. Verkley, 1995: Orography in a contour dynamics model of large-scale atmospheric flow. J. Atmos. Sci., 52 , 26432662.

    • Search Google Scholar
    • Export Citation
  • Appenzeller, C., , and H. C. Davies, 1992: Structure of stratospheric intrusions into the troposphere. Nature, 358 , 570572.

  • Bell, G. I., 1990: Interaction between vortices and waves in a simple model of geophysical flow. Phys. Fluids, 2A , 575586.

  • Blackmon, M. L., , J. E. Geisler, , and E. J. Pitcher, 1983: A general circulation model study of January climate anomaly patterns associated with interannual variation of equatorial Pacific sea surface temperature. J. Atmos. Sci., 40 , 14101425.

    • Search Google Scholar
    • Export Citation
  • Branstator, G., 1985a: Analysis of general circulation model sea surface temperature anomaly simulations using a linear-model. Part I: Forced solutions. J. Atmos. Sci., 42 , 22252241.

    • Search Google Scholar
    • Export Citation
  • Branstator, G., 1985b: Analysis of general-circulation model sea-surface temperature anomaly simulations using a linear-model. Part II: Eigenanalysis. J. Atmos. Sci., 42 , 22422254.

    • Search Google Scholar
    • Export Citation
  • Branstator, G., 2002: Circumglobal teleconnections, the jet stream waveguide, and the North Atlantic Oscillation. J. Climate, 15 , 18931910.

    • Search Google Scholar
    • Export Citation
  • Davies, H. C., , and C. H. Bishop, 1994: Eady edge waves and rapid development. J. Atmos. Sci., 51 , 19301946.

  • Davies, H. C., , and A. M. Rossa, 1998: PV frontogenesis and upper-tropospheric fronts. Mon. Wea. Rev., 126 , 15281539.

  • Fehlmann, R., , and H. C. Davies, 1999: Role of salient PV-elements in an event of frontal-wave cyclogenesis. Quart. J. Roy. Meteor. Soc., 125 , 18011824.

    • Search Google Scholar
    • Export Citation
  • Majewski, D., 1991: The Europa-Modell of the Deutscher Wetterdienst. Numerical Methods in Atmospheric Models, ECMWF Seminar Proceedings, Vol. 2, European Centre for Medium–Range Weather Forecasts, 147–191.

    • Search Google Scholar
    • Export Citation
  • Massacand, A. C., , and H. C. Davies, 2001: Interannual variability of the extratropical Northern Hemisphere and the potential vorticity wave guide. Atmos. Sci. Lett.,2, doi:10.1006/asle.2001. 0027.

    • Search Google Scholar
    • Export Citation
  • McIntyre, M. E., , and T. N. Palmer, 1984: The surf zone in the stratosphere. J. Atmos. Terr. Phys., 46 , 825849.

  • Meacham, S. P., , G. R. Flierl, , and U. Send, 1990: Vortices in shear. Dyn. Atmos. Oceans, 14 , 333386.

  • Nakamura, M., , and A. R. Plumb, 1994: The effects of flow asymmetry on the direction of Rossby wave breaking. J. Atmos. Sci., 51 , 20312045.

    • Search Google Scholar
    • Export Citation
  • Palmén, E., , and C. N. Newton, 1948: A study of the mean wind and temperature distribution in the vicinity of the polar front in winter. J. Meteor., 5 , 220226.

    • Search Google Scholar
    • Export Citation
  • Pedlosky, J., 1986: Geophysical Fluid Dynamics. 2d ed. Springer-Verlag, 710 pp.

  • Platzman, G. W., 1949: The motion of barotropic disturbances in the upper troposphere. Tellus, 1 , 5364.

  • Polvani, L. M., , and R. Plumb, 1992: Rossby wave breaking, microbreaking, filamentation, and secondary vortex formation: The dynamics of a perturbed vortex. J. Atmos. Sci., 49 , 462476.

    • Search Google Scholar
    • Export Citation
  • Pomroy, H. R., , and A. J. Thorpe, 2000: The evolution and dynamical role of reduced upper-tropospheric potential vorticity in intensive observing period one of FASTEX. Mon. Wea. Rev., 128 , 18171834.

    • Search Google Scholar
    • Export Citation
  • Pratt, L. J., , J. Earles, , P. Cornillon, , and J. F. Cayula, 1991: The non-linear behaviour of varicose disturbances in a simple model of the Gulf Stream. Deep-Sea Res., 38 , (Suppl. 1),. S591S622.

    • Search Google Scholar
    • Export Citation
  • Reed, R. J., , and E. F. Danielsen, 1960: Fronts in the vicinity of the tropopause. Arch. Meteor. Geophys. Bioklimatol., 11A , 117.

  • Schwierz, C., , and H. C. Davies, 2003: Evolution of a synoptic-scale vortex advecting toward a high mountain. Tellus, 55A , 158172.

  • Simmons, A. J., , J. M. Wallace, , and G. W. Branstator, 1983: Barotropic wave propagation and instability, and atmospheric teleconnection patterns. J. Atmos. Sci., 40 , 13631392.

    • Search Google Scholar
    • Export Citation
  • Sun, D. Z., , and R. S. Lindzen, 1994: A PV view of the zonal mean distribution of temperature and wind in the extratropical troposphere. J. Atmos. Sci., 51 , 757772.

    • Search Google Scholar
    • Export Citation
  • Swanson, K. L., , P. J. Kushner, , and I. M. Held, 1997: Dynamics of barotropic storm tracks. J. Atmos. Sci., 54 , 791810.

Fig. 1.
Fig. 1.

Instantaneous isentropic PV gradient on 1800 UTC 24 Dec 1999 (shaded, units 10−6 PVU m−1). (a) On the 320-K isentrope, 2-PVU PV contour and wind speed (contours 50, 70, 80, 90 m s−1) overlaid. (b) South–north cross section at 30°W. Vertical coordinate: θ; PV isolines [1, 2 (bold) 3, 4, 5, 6, 8 PVU]; and 50, 70, 80, 90 m s−1 isotachs (dashed) overlaid

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Fig. 2.
Fig. 2.

Mean winter (DJF) isentropic PV gradient for the 15-yr period 1979–93 (ERA-15 data). (a) On 320 K with wind speed (contours 30, 40, 50 m s−1). (b) Cross section at 75°W; 30 and 40 m s−1 isotachs (dashed) overlaid. Same as Fig. 1, but note the difference in scale and magnitude

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Fig. 3.
Fig. 3.

(a) Instantaneous PV on the 320-K isentrope at 1800 UTC 24 Dec 1999 (shaded, in PVU) with 2-PVU PV contour (bold black) and synoptic and mesoscale vortex anomalies (labels A–F) overlaid; (b) 15-yr winter (DJF) mean of negative vorticity on 500 hPa (shaded, units 10−4 s−1)

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Fig. 4.
Fig. 4.

Schematic diagram to illustrate the analytic model setup, terms, and symbols. (top) A vertical section at y = d. See text for details.

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Fig. 5.
Fig. 5.

(a) Initial velocity υa (dotted) and perturbation velocities after 2 (light) and 5 days (bold); units along x axis are 1000 km. (b) Amplitude evolution for different modes k in the domain of length Lϕ = 2πR cosϕ, with R (earth's radius) and Δk = 2π/Lϕ: 3Δk (dotted), Δk (bold) and 5Δk (light). Time in days. (c) Energy evolution [normalized, E(t)/E(0)] for the two extreme cases kr = 3Δk (bold) and kr = 3.5Δk (light). Also shown (dotted) is a function proportional to t2 (for t ≥ 10 days). (d) Spatial flow pattern after 5 days

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Fig. 6.
Fig. 6.

(a) Zonal velocity profile (solid line, units m s−1) and vorticity discontinuity [dashed, values 7.25 and 15 (×10−5 s−1)] of the barotropic initial state for the PE model operated on the sphere. The PV discontinuity is located at 47.5°N. (b) Initial positive vorticity anomaly (shaded, units 10−4 s−1) located north of the PV discontinuity (solid line).

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Fig. 7.
Fig. 7.

Time evolution (60–300 h) of the perturbation vorticity response (shaded, units 10−4 s−1) for a moving vorticity anomaly acting on a vorticity gradient at 47.5°N. Latitude circles shown for 40° and 80°N.

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Fig. 8.
Fig. 8.

Perturbation vorticity response at wavenumbers (top) m = 5, (bottom) m = 6, and (middle) for a nonresonant case for the integration times (left) 240 h and (right) 315 h. The different wavenumbers result from the particular location of the anomaly (bold contours 0.1, 0.3, 0.5 PVU) relative to the waveguide (65.5°, 63°, and 61.5°N, respectively). Cylindrical projection of the region 40°–75°N, 180°W–180°E.

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Fig. 9.
Fig. 9.

(a) Positive and (b) negative synoptic-scale vortex anomalies (a0 ∼ 700 km, lat0 ∼ 61°N) acting on a discontinuity at 47.5°N. Perturbation isentropic PV on 320 K for 0, 24, 36, 48, and 72 h. Contour interval is 0.15 PVU. Vortex anomaly amplitudes at initial time are 12 and −1.5 PVU, respectively

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Fig. 10.
Fig. 10.

Time evolution (60–300 h) of the perturbation vorticity response (shaded, units 10−4 s−1) for both an orographic stationary vorticity source (dashed contours) and a moving vorticity anomaly (solid contours) acting on a vorticity gradient at 47.5°N.

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Fig. 11.
Fig. 11.

Vorticity Hovmöller diagram at 47.5°N for a 480-h PE simulation (PV discontinuity located at 47.5°N, orography centered at 45°W)

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Save